On convex to pseudoconvex mappings

TL;DR

This paper characterizes self mappings of complex n-space that transform convex hypersurfaces into pseudoconvex ones, showing they are precisely those whose inverses are weakly pluriharmonic, satisfying a specific second order PDE.

Contribution

It provides a complete characterization of mappings transforming convex hypersurfaces into pseudoconvex ones via the weak pluriharmonicity of their inverses.

Findings

Mappings with weakly pluriharmonic inverses send convex to pseudoconvex hypersurfaces.

All pluriharmonic functions satisfy the key PDE, but other solutions exist.

The characterization links geometric properties to PDE conditions.

Abstract

In the works of Darboux and Walsh it was remarked that a one to one self mapping of $\rr^3$ which sends convex sets to convex ones is affine. It can be remarked also that a $\calc^2$-diffeomorphism $F:U\to U^{'}$ between two domains in $\cc^n$, $n\ge 2$, which sends pseudoconvex hypersurfaces to pseudoconvex ones is either holomorphic or antiholomorphic. \smallskip In this note we are interested in the self mappings of $\cc^n$ which send convex hypersurfaces to pseudoconvex ones. Their characterization is the following: {\it A $\calc^2$ - diffeomorphism $F:U'\to U$ (where $U', U\subset \cc^n$ are domains) sends convex hypersurfaces to pseudoconvex ones if and only if the inverse map $\Phi\deff F^{-1}$ is weakly pluriharmonic, i.e. it satisfies some nice second order PDE very close to $\d\bar\d \Phi = 0$.} In fact all pluriharmonic $\Phi$-s do satisfy this equation, but there are also other solutions.